A company called Argentia does an alarm that automatically triggers when the patient falls. You need to provide two contacts who could come out, but if they can't, then emergency services are contacted.

even if he won't wear an alarm, get your mum one so she can call for assistance with him easily when she needs it.

Have you had a social work assessment of your mothers needs? Sounds like she may be entitled to help for herself and if not is also entitled to a carers assessment. I'm not sure how it works if your Dad won't accept help for him but its worth a try.